The generator matrix

 1  0  1  1  1 3X+2  1  1  2  1  1 3X  1  1  0  1  1  2  1 3X+2  1  1  1 3X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0 2X  2  1  1  1  1 2X+2 3X+2 X+2  X  X
 0  1 X+1 3X+2 2X+3  1  2 X+3  1 3X 2X+1  1  0 X+1  1 3X+2 X+3  1 2X+3  1  2 3X 2X+1  1  0 3X+2 X+1 2X+1  2 X+3 2X+3 X+2 3X 2X 2X+2  X 3X+1  3 3X+3  1  0 X+2 2X X+2  1  1  1 2X  X  X 2X  1  1  1  2  2
 0  0 2X  0 2X  0 2X  0 2X 2X  0 2X  0  0  0 2X 2X 2X  0  0 2X  0 2X 2X 2X  0  0 2X  0  0 2X 2X  0 2X  0 2X 2X  0 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X  0
 0  0  0 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0 2X  0  0 2X 2X 2X  0  0 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0 2X  0  0 2X  0 2X  0 2X 2X  0  0 2X

generates a code of length 56 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+384x^54+266x^56+360x^58+3x^60+8x^62+1x^76+1x^80

The gray image is a code over GF(2) with n=448, k=10 and d=216.
This code was found by Heurico 1.16 in 17 seconds.